THE GAUSS CLASS NUMBER PROBLEM
FOR IMAGINARY QUADRATIC FIELDS
Dorian Goldfeld
§1. Introduction
√
Let D < 0 be a fundamental discriminant for an imaginary quadratic field K = Q( D). Such
fundamental discriminants D consist of all negative integers that are either ≡ 1 (mod 4) and
square–free, or of the form D = 4m with m ≡ 2 or 3 (mod 4) and square–free. We define
group of nonzero fractional ideals ba
h(D) = #
,
group of principal ideals (α), α ∈ K ×
to be the cardinality of the ideal class group of K. In the Disquisitiones Arithmeticae (1801) [G],
Gauss showed (using the language of binary quadratic forms) that h(D) is finite. He conjectured
that
h(D) −→ ∞ as D −→ −∞,
a result first proved by Heilbronn [H] in 1934. The Disquisitiones also contains tables of binary
quadratic forms with small class numbers (actually tables of imaginary quadratic fields of small
class number with even discriminant which is a much easier problem to deal with) and Gauss
conjectured that his tables were complete. In modern parlance, we can rewrite Gauss’ tables (we
are including both even and odd discriminants) in the following form.
h(D)
1
2
3
4
5
# of fields
9
18
16
54
25
largest |D|
163
427
907
1555
2683
The problem of finding an effective algorithm to determine all imaginary quadratic fields with
a given class number h is known as the Gauss class number h problem. The Gauss class number
problem is especially intriguing, because if such an effective algorithm did not exist, then the
associated Dirichlet L–function would have to have a real zero, and the generalized Riemann
hypothesis would necessarily be false. This problem has a long history (see [Go2]) which we do
not replicate here, but the first important milestones were obtained by Heegner [Heg], Stark [St1],
[St2], and Baker [B], whose work led to the solution of the class number one and two problems.
The general Gauss class number problem was finally solved completely by Goldfeld–Gross–Zagier
([Go1], [Go2], [G-Z]) in 1985. The key idea of the proof is based on the following theorem (see
[Go 1] (1976), for an essentially equivalent result) which reduced the problem to a finite amount
of computation.
This research is partially supported by the National Science Foundation.
Typeset by AMS-TEX
1
2
DORIAN GOLDFELD
Theorem 1: Let D be a fundamental discriminant of an imaginary quadratic field. If there exists
a modular elliptic curve E (defined over Q) whose associated base change Hasse–Weil L–function
LE/Q(√D) (s) has a zero of order ≥ 4 at s = 1 then for every > 0, there exists an effective
computable constant c (E) > 0, depending only on , E such that
h(D) > c (E)(log |D|)1− .
Note that the L–function of E/Q, LE (s), always divides LE/Q(√D) (s). If an imaginary quadratic
√
field Q( D) has small class number, then many small primes are inert. It is not hard to show
that the existence of an elliptic curve whose associated Hasse–Weil L–function has a triple zero at
s = 1 is enough to usually guarantee that LE/Q(√D) (s) has a fourth order zero. This idea will be
clarified in §3. We also remark, that if LE/Q(√D) (s) had a zero of order g ≥ 4, then you would get
(see [G1]) the lower bound
h(D) (log |D|)g−3 e−21
√
g(log log |D|)
.
Actually, [G1] also gives a similar result for real quadratic fields (D > 0),
h(D) log D (log |D|)g−3 e−21
√
g(log log |D|)
,
where D denotes the fundamental unit. In this case, however, it is required that LE/Q(√D) (s)
has a√ zero of order g ≥ 5 to get a non–trivial lower bound, because log D log D. The term
e−21 g(log log |D|) ( obtained by estimating a certain product of primes dividing D) is far from
optimal, because it simultaneously covers the cases of both real and imaginary quadratic fields. If
one considers only imaginary quadratic fields, the term can be easily written as a simple product
over primes dividing D. This was done by Oesterlé in 1985 [O] who made Theorem 1 explicit. He
proved that for (D, 5077) = 1,
1
h(D) >
log |D|
55
p|D, p=|D|
√ 2 p
1−
,
p+1
which allowed one to solve the class number 3 problem. More recently, using the above methods,
Arno [A] (1992), solved the class number four problem, and subsequently, work with Robinson
and Wheeler [A-R-W] (1998), and work of Wagner [Wag] (1996) gave a solution to Gauss’ class
number problem for class numbers 5, 6, 7. The most recent advance in this direction is due to
Watkins [Wat], who obtained the complete list of all imaginary quadratic fields with class number
≤ 100.
The main aim of this paper is to illustrate the key ideas of the proof of Theorem 1 by giving
full details of the proof for the solution of just the class number number one problem. The case
of class number one is considerably simpler than the general case, but the proof exemplifies the
ideas that work in general. We have not tried to compute or optimize constants, but have focused
instead on exposition of the key ideas.
Acknowledgment: The author would like to thank Brian Conrey for several helpful comments.
THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS
3
§2. The Deuring–Heilbronn Phenomenon
√
h(D) = 1. If a rational prime
Let Q( D) denote an imaginary quadratic field with class
number
√ √
m+n D
p splits completely in Q( D) then (p) = π · π̄ with π =
, a principal ideal. It follows
2
that
m2 − n 2 D
1 + |D|
p=
=⇒
p>
.
4
4
We have thus shown.
√
Lemma 2: Let Q( D) be an imaginary quadratic field of class number one. Then all primes less
must be inert.
than 1+|D|
4
Note that Lemma 2 can be used to write down prime producing polynomials [Ra]
x2 − x +
|D| + 1
,
4
(e.g., x2 − x + 41) which takes prime values for x = 1, 2, . . . , |D|−3
4 .
Lemma 2 is the simplest example of the more general phenomenon which says that an imaginary
quadratic field with small class number has the property that most small rational primes must be
inert in that field. It follows that if h(D) = 1, then the quadratic character χD (n) = D
n
√
(Kronecker symbol) associated to Q( D) satisfies χD (p) = −1 for most small primes, and thus
behaves like the Liouville function. Consequently, we heuristically expect that as D → −∞ and s
fixed with (s) > 12 ,
−1
χD (p)
L(s, χD ) =
1−
ps
p
−1
1
ζ(2s)
1+ s
,
∼
=
p
ζ(s)
p
√
so that analytically the Dirichlet L–function, L(s, χD ), associated to Q( D) behaves like ζ(2s)
ζ(s) . By
f (s) ∼ g(s) in a region s ∈ R ⊂ C we mean that there exists a small > 0 such that |f (s)−g(s)| < in the region R. Here we are appealing to the standard use of approximate functional equations
which allow one to replace an L–function by a short (square root of conductor) sum of its early
Dirichlet coefficients. This is the basis for the so called zero repelling effects (Deuring–Heilbronn
phenomenon) associated to imaginary quadratic fields with small class number. For example, if
h(D) = 1 and D → −∞, and D1 is a fixed discriminant of a quadratic field, then we expect that
for (s) > 12 ,
L(s, χD1 )L(s, χD χD1 ) ∼ L(2s, χD1 ),
which implies (see [Da]) that L(s, χD1 ) has no zeros γ + iρ with γ > 12 .
§3. Existence of L–functions of Elliptic Curves with Triple Zeroes
Let E be an elliptic curve defined over Q whose associated Hasse–Weil L–function LE (s) vanishes
at s = 1. Let LE (s, χd ) denote the L–function twisted by the quadratic character χd of conductor
4
DORIAN GOLDFELD
d, a fundamental discriminant of an imaginary quadratic field. We shall need the Gross–Zagier
formula (see [G–Z])
d
= cE Pd , Pd ,
LE (s)LE (s, χd )
(3.1)
ds
s=1
where Pd , Pd is the height pairing of a certain Heegner point PD and cE is an explicit constant
depending on the elliptic curve E. Gross and Zagier showed that if E is an elliptic curve of
conductor 37 and d = −139, then the Heegner point is torsion and the height pairing Pd , Pd vanishes. By (3.1), this gives a construction of an L–function with a triple zero at s = 1. Actually,
their method is quite general, and many other such examples can be constructed.
Henceforth, we fix E to be the above elliptic curve of conductor N = 37 · 1392 . Then the
Hasse–Weil L–function LE (s) satisfies the functional equation (see [Shim])
√ 1−s
√ 1+s
N
N
Γ(1 + s)LE (1 + s) = −
Γ(1 − s)LE (1 − s),
2π
2π
and LE (1 + s) has a MacLaurin expansion of the form
LE (1 + s) = c3 s3 + c4 s5 + {higher odd powers of s}.
Now, let D, with |D| > 163, denote a fundamental discriminant of an imaginary quadratic field
with class number one. It is not hard to show that (D, 37 · 139) = 1. Let χD denote the quadratic
Dirichlet character of conductor D. We define
s
N |D|
(3.2)
ΛD (s) =
Γ(1 + s)2 LE (s)LE (s, χD ).
4π 2
Then it can be shown (see [Shim]) that ΛD (s) satisfies the functional equation
(3.3)
ΛD (1 + s) = w · ΛD (1 − s),
with root number √
w = χD (−37 · 1392 ) = χD (−37) = +1, because the early primes of an imaginary
quadratic field Q( D) with class number one must be inert (Lemma 2). It follows from (3.3) that
LE/Q(√D) (s) = LE (s)LE (s, χD )
has a zero of even order at s = 1. Since LE (s) has a zero of order 3 at s = 1, we immediately see
that LE/Q(√D) (s) must have a zero of order at least 4 at s = 1. This is the main requirement of
Theorem 1.
§4. Solution of the Class Number One Problem
√
Assume D is sufficiently large and the class number h(D) of Q( D) is one. We will get a
contradiction using zero–repelling ideas (Deuring–Heilbronn phenomenon) of section 2. The main
idea is to consider the integral ID defined by:
ID
1
=
2πi
2+i∞
ΛD (1 + s)
2−i∞
ds
,
s3
THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS
5
where ΛD (1 + s) is given in (3.2).
Lemma 3: We have ID = 0.
Proof: If we shift the line of integration to (s) = −2, the residue at s = 0 is zero because
ΛD (1 + s) has a fourth order zero at s = 0. If immediately follows that
ID
1
=
2πi
−2+i∞
ΛD (1 + s)
−2−i∞
1
=−
2πi
2+i∞
ΛD (1 + s)
ds
s3
ds
s3
2−i∞
= −ID ,
after applying the functional equation (3.3) and letting s → −s. Consequently, ID = 0. This
completes the proof of Lemma 3.
We will now show that if h(D) = 1 and D is sufficiently large then ID = 0. The heuristics for
obtaining this contradiction are easily seen. We may write the Euler products:
(4.1)
LE (s) =
p
LE (s, χD ) =
1−
p
αp
1− s
p
−1 αp χD (p)
ps
βp
1− s
p
−1 1−
−1
,
βp χD (p)
ps
−1
.
The assumption that h(D) = 1 implies that χD (p) = −1 for all primes p < 1+|D|
(Lemma 2). So
4
we expect that analytically the Euler product LE (s)LE (s, χD ) should behave like
φ(s) :=
p
αp2
1 − 2s
p
−1 βp2
1 − 2s
p
−1
,
where
|αp |2 = |βp |2 = αp βp = p,
for all but finitely many primes p. Now, if f is the weight two Hecke eigenform associated to E,
then we have the symmetric square L–function
−1 −1
−1 2
2
α
β
α
β
p p
p
p
1− s
1− s
1− s
.
L s, sym2 (f ) :=
p
p
p
p
2
2
(f
)
/ζ(2s
−
1).
It
is
known
that
L
s,
sym
(f
)
is entire which
Thus φ(s) is essentially
L
2s,
sym
2
implies that L 2s, sym (f ) /ζ(2s − 1) vanishes at s = 1, a result first proved by [Ogg]. This
implies that
φ(1) = 0.
6
DORIAN GOLDFELD
In fact, φ(1) has a simple zero at s = 1 which seems to contradict the fact that LE (s)LE (s, χD ) has
a fourth order zero. Although it appears that a contradiction could be obtained if LE (s)LE (s, χD )
had a double zero at s = 1, this, unfortunately is not the case. The contradiction is much more
subtle and will be shortly clarified.
We now define
∗
ID
2+i∞
1
=
2πi
37 · 1392 |D|
4π 2
1+s
Γ(1 + s)2 φ(1 + s)
ds
,
s3
2−i∞
which allows us to write
∗
+ Error,
0 = ID = ID
(4.2)
with
(4.3)
1
Error =
2πi
2+i∞
37 · 1392 |D|
4π 2
1+s
ds
Γ(1 + s)2 LE (1 + s)LE (1 + s, χD ) − φ(1 + s) 3 .
s
2−i∞
Lemma 4: Define Dirichlet coefficients Bn (n = 1, 2, . . . ) by the representation
LE (1 + s)LE (1 + s, χD ) − φ(1 + s) =
∞
Bn n−1−s .
n=1
We also define Dirichlet coefficients νD (n)(n = 1, 2, . . . ) by the representation
ζ(s)(L(s, χD ) =
∞
νD (n) n−s .
n=1
Then Bn = 0 for n <
1+|D|
4 .
In the other cases, we have
|Bn | ≤
where d4 (n) =

√

 2νD (n) n

 2d (n) · √n
4
2
≤ n < 1+|D|
,
4
2
if n ≥ 1+|D|
,
4
if
1+|D|
4
1.
d1 d2 d3 d4 =n
Proof: The fact that Bn = 0 for n < 1+|D|
follows immediately from Lemma 2. The upper
4
bound |Bn | ≤ 2d4 (n) · n is a consequence of the fact (see (4.1)) that LE (1 + s)LE (1 + s, χD ) is an
Euler product of degree 4. Thus, the Dirichlet coefficients of LE (1 + s)LE (1 + s, χD ) are bounded
by the Dirichlet coefficients of the Euler product
p
√ −1 ∞
√
p
=
d4 (n) n · n−1−s .
1 − 1+s
p
n=1
THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS
7
The extra factor of 2 in the bound for Bn comes from the consideration of the additional Euler
product for φ(1 + s).
2
1+|D|
≤
n
<
, we can only have Bn = 0 if n is divisible by a prime
In the range 1+|D|
4
4
2
q > 1+|D|
4 . In this range, it is not possible that q divides n. This implies that φ(1 + s) does not
contribute to Bn since φ(1 + s) is a Dirichlet series formed from perfect squares, i.e., of the form
∞
∞
b(k)
. If we let n = q·m then we must have Bn = am ·aq where LE (s) =
ak ·k −s .
φ(1+s) =
(k2 )1+s
k=1
k=1
√
Consequently, |Bn | ≤ 2|am | q. It is easy to see that m must be a perfect square because m can
only be divisible by primes < 1+|D|
4 . Again, by considering the Euler product (4.1), it follows that
2
√
√
in the range 1+|D|
≤ n < 1+|D|
, the coefficients Bn are bounded by 2νD (n) n where νD (n) n
4
4
are the Dirichlet coefficients of the Euler product
p
√ −1 √ −1 ∞
√
p
p
=
νD (n) n · n−1−s .
1 − 1+s
1 − χD (p) 1+s
p
p
n=1
Clearly,
ζ(s)L(s, χD ) =
∞
νD (n) · n−s .
n=1
Lemma 5: Let x > 1. Then
√
3
3
1
νD (n) n ≤ 4e · x 2 L(1, χD ) + O |D| 2 x− 2 .
x≤n≤2x
If we further assume that |D| > 4 and h(D) = 1, then
√
3
1
x2
2 x− 2
νD (n) n ≤ 4πe ·
+
O
|D|
.
1
2
|D|
x≤n≤2x
3
Proof: We shall need the well known Mellin transform:
2+i∞
1
2πi
xs Γ(s) ds = e− x .
1
2−i∞
It follows that
x≤n≤2x
√
2e
νD (n) n ≤
2πi
= 2e
2+i∞
1
ζ s−
2
2−i∞
∞
1
(2x)s − xs Γ(s) ds
L( s − , χD
2
√ n
n
νD (n) n e− 2x − e− x .
n=1
8
DORIAN GOLDFELD
n
n
Here we have used the fact that 2e e− 2x − e− x > 1 for x ≤ n ≤ 2x, and, otherwise, νD (n) ≥ 0.
The above integral can be evaluating by shifting the line of integration to the left to the line
(s) = − 12 . There is a pole at s = 32 coming from the Riemann zeta function. Consequently
(4.4)
√
3
3
νD (n) n ≤ 2eL(1, χD ) (2x) 2 − x 2
x≤n≤2x
− 12 +i∞ 2e
1
1
+ (2x)s − xs Γ(s) ds .
L( s − , χD
ζ s−
2πi
2
2
− 1 −i∞
2
The functional equation
|D|
π
ζ(s)L(s, χD ) =
1−2s
2−s Γ 1−s
Γ
2
s 1+s2 ζ(1 − s)L(1 − s, χD )
Γ 2 Γ 2
together with Stirling’s asymptotic formula
lim |Γ(σ + it)|e 2 |t| |t| 2 −σ =
π
1
√
|t|→∞
2π
3
imply that the shifted integral in (4.4) converges absolutely and is bounded by O |D| 2 x−1 . This
completes the first part of the proof of Lemma 5. For the second part, we simply use Dirichlet’s
class number formula (see [Da]), L(1, χD ) = πh(D)
1 , which holds for |D| > 4.
|D| 2
Lemma 6: For y > 0, define
2+i∞
1
G(y) :=
2πi
y s+1 Γ(1 + s)2
ds
.
s3
2−i∞
Then
− √1y
G(y) < 2y 2 e
.
Proof: Recall the definition of the Gamma function
∞
du
Γ(s) =
,
e−u us
u
0
which satisfies Γ(s + 1) = sΓ(s). It follows that
(4.5)
1
G(y) =
2πi
2+i∞
y
2−i∞
s+1
0
∞
0
∞
e−u1 −u2 (u1 u2 )s
du1 du2 ds
.
u1 u2 s
THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS
On the other hand, we have the classical integral
1
2πi


1
2+i∞
ds
=
x

s

s
2−i∞
9
if x > 1,
1
2
if x = 1,
0
if x < 1.
If we now apply the above to (4.5) (after interchanging integrals), we obtain
du1 du2
e−u1 −u2
.
(4.6)
G(y) = y
u1 u2
u1 u2 ≥ y −1
To complete the proof, we use the range of integration, u1 u2 ≥ y −1 , to show that
which it follows from (4.6) that
2
e−u1 −u2 du1 du2
G(y) ≤ y
= y
u1 u2 ≥ y −1
∞
− u1 y −u2
2
e
0
3
∞
= y2
≤ 2y
3
2
− √1y u2 + u1
du2
− √1y u2 + u1
e
1
1
2 − √y
< 2y e
2
∞
≤ y, from
du2
2
e
0
1
u1 u2
2
du2
.
It now follows from (4.3), Lemma 4, and the definition of G(y) given in Lemma 6 that
37 · 1392 |D|
|Error| ≤
.
|Bn | · G
4π 2 n
1+|D|
n≥
4
The bound for G(y) given in Lemma 6 implies that
|Error| ≤
1+|D|
≤n≤
4
(
√
4νD (n) n ·
1+|D|
4
2
2
( 1+|D|
)
4
37 · 1392 |D|
4π 2 n
√
4d4 (n) n ·
37 · 1392 |D|
4π 2 n
4π 2 n
− 37·139
2 |D|
2
)
+
2
e
4π 2 n
− 37·139
2 |D|
e
.
<n
√ The second sum in the above Error is O e−c1 |D| (for some c1 > 0), so can be ignored. We can,
therefore, estimate the Error by breaking it into smaller sums as follows:
|Error| ≤ 4
k<log2 (
1+|D|
4
)
372 · 1394
22k−2 · π 4
√
4π 2 n
− 37·139
2 |D|
νD (n) n · e
1+|D| k−1
2
≤n≤
4
(
1+|D|
4
)2k
√ + O e− |D| .
10
DORIAN GOLDFELD
For each, k, we can apply Lemma 5 to the inner sum over n in the above. It follows that
k
|Error| |D|
2− 2 |D|.
k<log(
1+|D|
4
)
It immediately follows that for D sufficiently large, there exists a fixed, effectively computable
constant c such that
|Error| ≤ c · |D|
as |D| → ∞. Combining this bound with (4.2), we have that
(4.7)
∗
ID
1
=
2πi
2+i∞
37 · 1392 |D|
4π 2
1+s
Γ(1 + s)2 φ(1 + s)
ds
s3
2−i∞
satisfies
(4.8)
∗
|ID
| < c · |D|.
∗
The integral for ID
given in (4.7) can be evaluated by shifting the line of integration to the left.
A double pole is encountered at s = 0. Actually the term 1/s3 contributes a triple pole, but the
vanishing of φ(1 + s) at s = 0 reduces this to a double pole. Because of the double pole and the
known zero–free region for the Riemann zeta function, it is not hard to show that there exists an
effectively computable constant c1 > 0 such that
(4.9)
∗
|ID
| > c1 D log D.
The inequalities (4.8) and (4.9) are contradictory for large D. Consequently, it is not possible that
h(D) = 1. QED
§5. References
[A] S. Arno, The imaginary quadratic fields of class number 4, Acta Arith. 60 (1992), 321–334.
[A-R-W] S. Arno, M. Robinson, F. Wheeler, Imaginary quadratic fields with small odd class
number, Acta Arith. 83 (1998), 295–330.
[B] A. Baker, Imaginary quadratic fields with class number 2, Annals of Math. (2) 94 (1971),
139–152.
[Da] H. Davenport, Multiplicative Number Theory, Second edition, Revised by H. Montgomery,
Grad. Texts in Math. 74, Springer–Verlag (1980).
[G] C.F. Gauss, Disquisitiones Arithmeticae, Göttengen (1801); English translation by A. Clarke,
revised by W. Waterhouse, 1986 Springer–Verlag reprint of the Yale University Press, New Haven,
1966 edition.
[Go1] D. Goldfeld The class number of quadratic fields and the conjectures of Birch and Swinnerton–
Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 624–663.
THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS
11
[Go2] D. Goldfeld, Gauss’ class number problem for imaginary quadratic fields, Bull. Amer. Math.
Soc. 13 (1985), 23–37.
[G-Z] B. Gross, D.B. Zagier, Heegner points and derivatives of L–series, Invent. Math. 84 (1986),
225–320.
[Heg] K. Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253.
[H] H. Heilbronn, On the class number in imaginary quadratic fields, Quarterly J. of Math., 5
(1934), 150–160.
[O] J. Oesterlé, Le probléme de Gauss sur le nombre de classes, Enseign. Math. 34 (1988), 43–67.
[Ogg] A. Ogg, On a convolution of L–series, Invent. Math. 7 (1969), 297–312.
[Ra] G. Rabinovitch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern,
Proc. Fifth Internat. Congress Math. (Cambridge), vol I, (1913), 418–421.
[Shim] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami
Shoten – Princeton University Press, (1971).
[St1] H. Stark, A complete determination of the complex quadratic fields of class–number one,
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[St2] H. Stark, A transcendence theorem for class–number problems I, II, Annals of Math. (2) 94
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[Wag] C. Wagner, Class number 5, 6 and 7, Math. Comp. 65 (1996), 785–800.
[Wat] M. Watkins, Class numbers of imaginary quadratic fields, to appear.
Dorian Goldfeld, Columbia University Department of Mathematics, New York, NY 10027
E-mail address: [email protected]